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Let $\Gamma({\mathcal{W}})$ be the Cayley graph of a finite Weyl groupoid ${\mathcal{W}}$. In this paper, we show an existence of a Hamitonian cycle of $\Gamma({\mathcal{W}})$ for any ${\mathcal{W}}$. We exatctly draw a Hamiltonian cycle of $\Gamma({\mathcal{W}})$ for any (resp. some) irreducible ${\mathcal{W}}$ of rank three (resp. four). Moreover for the irreducible ${\mathcal{W}}$ of rank three, we give a second largest eigenvalue of the adjacency matrix of $\Gamma({\mathcal{W}})$, and know if $\Gamma({\mathcal{W}})$ is a bipartite Ramanujan graph or not., Comment: 38 pages, 6 figures, any comment is welcome

We study Hamilton circuits of the Cayley graphs of the Weyl groupoids of the generalized quantum groups, or the quantum double of the Nichols algebras of diagonal-type, with finite root systems. We prove the existence of a Hamilton circuit for any of them and explicitly draw one of them for rank 3 and 4 cases., Comment: 56 pages, 71 figures

Ge atoms segregating on zirconium diboride thin films grown on Ge(111) were found to crystallize into a two-dimensional bitriangular structure which was recently predicted to be a flat band material. Angle-resolved photoemission experiments together with theoretical calculations verified the existence of a nearly flat band in spite of non-negligible in-plane long-range hopping and interactions with the substrate. This provides the first experimental evidence that a flat band can emerge from the electronic coupling between atoms and not from the geometry of the atomic structure.

We introduce the definition of the typical irreducible modules of the generalized quantum groups, and prove the Weyl-Kac-type formulas of their characters. As a by-product, we obtain the Weyl-Kac-type character formulas of the typical irreducible modules of the quantum superalgebras associated with the basic classical Lie superalgebras, which is explained in Introduction., Comment: 28 pages

This paper gives elements in the (skew) center of the generalized quantum group corresponding to its irreducible finite dimensional modules. Finally we give a conjecture stating that those must form a basis of the center., Comment: 17 pages

Let $\pi:sl(n|n)\to A(n-1,n-1)$ be the natural epimorphism of Lie superalgebra. Then $\dim\ker\pi=1$. Let $\pi^{(t)}:sl^{(t)}(n|n)\to A^{(t)}(n-1,n-1)$ be the natural epimorphism, where $t=1,2,4$. Let $\{e_k|k\in{\mathbb{Z}}\}$ be the basis of $\ker\pi^{(t)}$ with $e_k\in sl^{(t)}(n|n)_{(a_tk+b_t)\delta}$, where $(a_1,b_1)=(1,0)$, $(a_2,b_2)=(2,-1)$ and $(a_4,b_4)=(4,-2)$. The main result of this paper is to explicitly describe an element of $U_q(sl^{(t)}(n|n))$ (and its multi-parameter version) corresponding to $e_1$ (i.e., $k=1$). As for $U_q(sl^{(1)}(n|n))$ (i.e., $t=1$), the author had already had explicit description for every $k$ in 1999., Comment: 13 pages, 3 figures, submitted to Proceedings of Meeting for Study of Number theory, Hopf algebras and related topics, 2017 February 12-15, the room B121 of Faculty of Science, University of Toyama, Japan

We study the Kostant-Lusztig $\mathbb A$-base of the multiparameter quantum groups. To simplify calculations, especially for $G_2$-type, we utilize the duality of the pairing of the universal $R$-matrix., Comment: Submitted to Contemporary Math. on July 21, 2017

We introduce nil-Hecke algebras for Weyl groupoids. We describe a basis and some properties of these algebras which lead to a notion of Bruhat order for Weyl groupoids., Comment: 13 pages

A 1D metallic surface state was created on an anisotropic InSb(001) surface covered with Bi. Angle-resolved photoelectron spectroscopy (ARPES) showed a 1D Fermi contour with almost no 2D distortion. Close to the Fermi level ($E_{\rm F}$), the angle-integrated photoelectron spectra showed power-law scaling with the binding energy and temperature. The ARPES plot above $E_{\rm F}$ obtained thanks to thermally broadened Fermi edge at room temperature showed a 1D state with continuous metallic dispersion across $E_{\rm F}$ and power-law intensity suppression around $E_{\rm F}$. These results strongly suggest a Tomonaga-Luttinger liquid on the Bi/InSb(001) surface., Comment: 12 pages (4 figures) for main text, 5 pages (6 figures) for supplemental material

This paper treats the generalized quantum group $U=U(\chi,\pi)$ with a bi-homomorphism $\chi$ for which the corresponding generalized root system is a finite set. We establish a Harish-Chandra type theorem describing the (skew) center of $U$., Comment: Some comments have been added. 49pages, accepted for publication in Journal of Pure and Applied Algebra

Let $H$ be the quantum double of a Nichols algebra of diagonal type. We compute the $R$-matrix of 3-uples of modules for general finite-dimensional highest weight modules over $H$. We calculate also a multiplicative formula for the universal $R$-matrix when $H$ is finite dimensional., Comment: 22 pages

We investigated Pt-induced nanowires on the Si(110) surface using scanning tunneling microscopy (STM) and angle-resolved photoemission (ARP). High resolution STM images show a well-ordered nanowire array of 1.6 nm width and 2.7 nm separation. ARP reveals fully occupied one dimensional (1D) bands with a Rashba-type split dispersion. Local dI/dV spectra further indicate well confined 1D electron channels on the nanowires, whose density of states characteristics are consistent with the Rashba-type band splitting. The observed energy and momentum splitting of the bands are among the largest ever reported for Rashba systems, suggesting the Pt-Si nanowire as a unique 1D giant Rashba system. This self-assembled nanowire can be exploited for silicon-based spintronics devices as well as the quest for Majorana Fermions., Comment: to be published in Phys. Rev. Lett

The electronic structure of the unoccupied and occupied states of potassium (K)-doped and undoped picene crystalline films has been investigated by using the element-selective and bulk-sensitive photon-detection methods of X-ray absorption and emission spectroscopies. We observed the formation of the doping-induced unoccupied and occupied electronic states in K-doped picene. By applying the inner-shell resonant-excitation experiments, we observed the evidence for the orbital hybridization between K and picene near the Fermi energy. Furthermore, the resonant X-ray emission experiment suggests the presence of the Raman-active vibronic interaction in K-doped picene. These experimental evidences play a crucial role in the superconductivity of K-doped picene., Comment: 12 pages, 3 figures

We classify finite-dimensional irreducible highest weight modules of generalized quantum groups whose positive part is infinite dimensional and has a Kharchenko's PBW basis with an irreducible finite positive root system., Comment: 59 pages, We give and prove some reduced expressions of the longest elements of not only the Weyl groupoids treated in the paper, but also the finite Weyl groups

The notion of a "root base" together with its geometry plays a crucial role in the theory of finite and affine Lie theory. However, it is known that such a notion does not exist for the recent generalizations of finite and affine root systems such as extended affine root systems and affine reflection systems. As an alternative, we introduce the notion of a "reflectable base", a minimal subset $\Pi$ of roots such that the non-isotropic part of the root system can be recovered by reflecting roots of $\Pi$ relative to the hyperplanes determined by $\Pi$. We give a full characterization of reflectable bases for tame irreducible affine reflection systems of reduced types, excluding types $E_{6,7,8}$. As a byproduct of our results, we show that if the root system under consideration is locally finite then any reflectable base is an integral base., Comment: 33 pages, some minor changes have been done. Also some references are updated

We discuss the relationship between Hopf superalgebras and Hopf algebras. We list the braided vector spaces of diagonal type with generalized root system of super type and give the defining relations of the corresponding Nichols algebras., Comment: 18 pages

This is an exposition in order to give an explicit way to understand (1) a non-topological proof for an existence of a base of an affine root system, (2) a Serre-type definition of an elliptic Lie algebra with rank =>2, and (3) the isotropic root multiplicities of those elliptic Lie algebras., Comment: Submitted to RIMS Kokyuroku Bessatsu

Using the well-known recognition and structural theorem(s) for root-graded Lie algebras and their universal coverings, we give a finite presentation for the universal covering algebra of a centerless Lie torus of type $X\not=A,C,BC$. We follow a unified approach for the types under consideration.

In this paper we give Iwahori-Hecke type algebras H_q(g) associated with the Lie superalgebras g=A(m,n), B(m,n), C(n) and D(m,n). We classify the irreducible representations of H_q(g) for generic q., Comment: Accepted for publication in RIMS Kokyuroku Bessatsu

In this paper, we give a finite number of defining relations satisfied by a finite number of generators for the elliptic Lie algebras and superalgebras ${\frak g}_R$ with rank $\geq 2$. Here the $R$'s denote the reduced and non-reduced elliptic root systems with rank $\geq 2$. We also show that if ${\cal L}$ is an extended affine Lie algebra (EALA) whose non-isotropic roots form the $R$, then there exists a natural homomorphism ${\cal F}:{\frak g}_R \to{\cal L}$, which also give a universal central extension (UCE) surjective map from $[{\frak g}_R,{\frak g}_R]$ to the core of ${\cal L}$. (More precisely, we take a ${\bar {\frak g}}_R$ instead of the ${\frak g}_R$.), Comment: 30 pages

In this paper, using a quantum superalgebra associated with the universal central extension of sl(2,2)^{(1)}, we introduce new R-matrices having an extra parameter x. As x\to 0, they become those associated with the symmetric and anti-symmetric tensor products of the copies of the vector representation of U_q sl(2,2)^{(1)}., Comment: 18 pages, latex, v2.2 references added

This paper has been withdrawn by the author because he was informed of the following paper: Cummins C. and King R. C., An algorithm for calculating characters of Hecke algebras Hn (q) of tyoe An-1 when q is a root of unity, Comm. Algebra 21 (1993), no.12 4425-4437 by its first author. The author's main result seems to be included in their result., Comment: Paper withdrawn

In this paper, we give defining relations of the affine Lie superalgebras an and defining relations of a super-version of the Drinfeld[D]-Jimbo[J] affine quantized universal enveloping algebras. As a result, we can exactly define the affine quantized universal enveloping superalgebras with generators and relations. Moreover we give a Drinfeld's realizations of $U_h({\hat {sl}}(m|n)^{(1)})$., Comment: latex, 29k bytes